A colleague of yours is faced with testing a relationship at the .05 level or...

Level of significance
The level of significance is defined as the probability of rejecting a null hypothesis by the test when it is really true, which is denoted as $\alpha$.
In hypothesis testing, greater the significance level greater the chance to reject the null hypothesis.
If we choose 0.05 level of significance, then there may greater chance to reject the null hypothesis.
Therefore, the advice given to the colleague with testing a relationship to select the 0.05 level of significance for testing a relationship.